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Points and Lines : Characterizing the Classical Geometries / by Ernest Shult
Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Universitext | SpringerLink BücherPublisher: Berlin, Heidelberg : Springer-Verlag Berlin Heidelberg, 2011Description: Online-Ressource (XXII, 676p. 88 illus, digital)ISBN:- 9783642156274
- 516
- QA440-699
Contents:
Summary: I.Basics -- 1 Basics about Graphs -- 2 .Geometries: Basic Concepts -- 3 .Point-line Geometries.-4.Hyperplanes, Embeddings and Teirlinck's Eheory -- II.The Classical Geometries -- 5 .Projective Planes.-6.Projective Spaces -- 7.Polar Spaces -- 8.Near Polygons -- III.Methodology -- 9.Chamber Systems and Buildings -- 10.2-Covers of Chamber Systems -- 11.Locally Truncated Diagram Geometries.-12.Separated Systems of Singular Spaces -- 13 Cooperstein's Theory of Symplecta and Parapolar Spaces -- IV.Applications to Other Lie Incidence Geometries -- 15.Characterizing the Classical Strong Parapolar Spaces: The Cohen-Cooperstein Theory Revisited -- 16.Characterizing Strong Parapolar Spaces by the Relation between Points and Certain Maximal Singular Subspaces -- 17.Point-line Characterizations of the “Long Root Geometries” -- 18.The Peculiar Pentagon Property.Summary: The classical geometries of points and lines include not only the projective and polar spaces, but similar truncations of geometries naturally arising from the groups of Lie type. Virtually all of these geometries (or homomorphic images of them) are characterized in this book by simple local axioms on points and lines. Simple point-line characterizations of Lie incidence geometries allow one to recognize Lie incidence geometries and their automorphism groups. These tools could be useful in shortening the enormously lengthy classification of finite simple groups. Similarly, recognizing ruled manifolds by axioms on light trajectories offers a way for a physicist to recognize the action of a Lie group in a context where it is not clear what Hamiltonians or Casimir operators are involved. The presentation is self-contained in the sense that proofs proceed step-by-step from elementary first principals without further appeal to outside results. Several chapters have new heretofore unpublished research results. On the other hand, certain groups of chapters would make good graduate courses. All but one chapter provide exercises for either use in such a course, or to elicit new research directions.PPN: PPN: 1650743971Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
""Preface""; ""Contents""; ""Part I Basics""; ""1 Basics About Graphs ""; ""1.1 The Language of Graphs""; ""1.1.1 Connectedness and the Distance Metric""; ""1.1.2 Subgraphs""; ""1.1.3 Special Types of Graphs and Subgraphs""; ""1.1.4 Metrical Properties of Subgraphs""; ""1.2 Morphisms of Graphs""; ""1.2.1 Basic Definitions""; ""1.2.2 Fiberings, Covering Morphisms, and Lifts of Walks""; ""1.2.3 Universal C-Covers""; ""1.3 C-Homotopy""; ""1.3.1 Further Properties of the Closure Operator on Circuits""; ""1.3.2 Control of C-Connectedness Through a Subgraph""
""1.3.3 Tits' Condition for Being Simply C-Connected""""1.4 The Existence of Universal C-Covers""; ""1.4.1 C-Connectedness and Subgraphs""; ""1.4.2 The Construction""; ""1.4.3 Deck Transformations""; ""1.5 Exercises for Chapter 1""; ""1.5.1 Exercises for Section 1.1""; ""1.5.2 Exercises for Section 1.2""; ""1.5.3 Exercises for Section 1.3""; ""2 Geometries: Basic Concepts ""; ""2.1 Introduction""; ""2.2 Geometries: Definitions and Basic Concepts""; ""2.2.1 Basic Definitions""; ""2.2.2 Subgeometries""; ""2.2.3 Truncations""; ""2.2.4 Point-Line Geometries""
""2.2.5 Flags and Chambers""""2.2.6 Residues""; ""2.2.7 The Interplay of Residues and Truncations""; ""2.2.8 Shadows""; ""2.3 Examples""; ""2.4 Morphisms of Geometries""; ""2.4.1 Definition""; ""2.4.2 Automorphisms""; ""2.4.3 Morphisms Defined by a Group of Automorphisms""; ""2.4.4 Truncations and Morphisms""; ""2.4.5 Residues and Morphisms""; ""2.5 Connectedness Properties""; ""2.5.1 Residual Connectedness""; ""2.6 Exercises for Chapter 2""; ""3 Point-Line Geometries ""; ""3.1 Introduction""; ""3.1.1 On Choosing a Reasonable Definition""
""3.1.2 Our Definition of Point-Line Geometry""""3.2 The Point-Collinearity Graph""; ""3.3 Morphisms and Covers of Point-Line Geometries""; ""3.4 Subspaces""; ""3.4.1 Generalizations of the Notion of Subspace""; ""3.5 Special Types of Point-Line Geometries""; ""3.5.1 Partial Linear Spaces and Linear Spaces""; ""3.5.2 Gamma Spaces""; ""3.6 Local Connectedness in Gamma Spaces""; ""3.6.1 The Decomposition of a Gamma Space into Locally Connected Components""; ""3.6.2 How Local Characterizations of Gamma Spaces Reduce to the Locally Connected Case""; ""3.7 Enriching Geometries""
""3.8 Products of Point-Line Geometries, a Construction""""3.9 Exercises and Examples for Chapter 3""; ""4 Hyperplanes, Embeddings, and Teirlinck's Theory ""; ""4.1 Veldkamp Spaces""; ""4.1.1 Introduction""; ""4.1.2 Geometric Hyperplanes""; ""4.1.3 The Veldkamp Space""; ""4.2 Teirlinck's Theory""; ""4.2.1 The Hypotheses""; ""4.2.2 The Exchange Property in Closed Sets""; ""4.2.3 To What Extent Does H Separate Points?""; ""4.2.4 The H-Closure of Two Inequivalent Points""; ""4.2.5 The Natural Morphism""; ""4.2.6 Singular Subspaces are Generalized Projective Spaces""
""4.3 The Effect of Teirlinck's Theory on the Veldkamp Space""
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